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X(4), X(98), X(112), X(511), X(1113), X(1114), X(2592), X(2593) imaginary foci of the Brocard inellipse, on the Brocard axis and on the Kiepert hyperbola other points below 

Geometric properties : 

The polar conic of X(4) in any cubic of the Euler pencil in Table 27 is a rectangular hyperbola. These belong to a same pencil and have four common points X(4) and H1, H2, H3 forming an orthocentric system. The polar lines of any point M in all these hyperbolas concur at M* which is the isogonal conjugate of M with respect to the diagonal triangle P1P2P3 of the system. It follows that X(4), H1, H2, H3 are the in/excenters of P1P2P3. See Table 62 when polar conics of X(3) are considered. Now, for any P, the locus of M such that P, M, M* are collinear is a cubic which is obviously a pK with respect to P1P2P3. When P is on the line at infinity, the cubic is circular and the most interesting example is probably K1096 with P = X(511).

(J) is the Jerabek hyperbola, the polar conic of X(4) in K003. Its remaining common points with K1096 are X(32618) and X(32619), the isogonal conjugates of X(5000) and X(5001) respectively. These two points lie on the line passing through X(2), X(98). (D) is a diagonal conic in ABC, the polar conic of X(4) in K006. It passes through X(1), X(4), X(19), X(920), X(1075), X(1249), X(1712), X(1713), X(1714), X(1715), X(2588), X(2589), X(3068), X(3069), X(3183), X(3186), etc. (H) is the polar conic of X(511) in K1096. It passes through X(4), X(39), X(511), X(512), X(2211), etc, and also H1, H2, H3. (H) is also the polar conic of X(4) in pK(X6, E) where E = a^2 (a^8 b^23 a^6 b^4+3 a^4 b^6a^2 b^8+a^8 c^2+a^6 b^2 c^2+4 a^4 b^4 c^23 a^2 b^6 c^23 b^8 c^23 a^6 c^4+4 a^4 b^2 c^4+3 b^6 c^4+3 a^4 c^63 a^2 b^2 c^6+3 b^4 c^6a^2 c^83 b^2 c^8) : : , on the Euler line, SEARCH = 51.4695540486639. (C) is the circumcircle of P1P2P3, the locus of centers of all rectangular hyperbolas. (C) is orthogonal to the circumcircle (O) and to the nine point circle (N). It passes through X(98), X(107), X(125), X(132), X(5000), X(5001), etc, and its center is X(6130). F is the singular focus of K1096, the reflection of X(98) about X(6130). It lies on (C) and on the line X(99)X(107). SEARCH = 31.8400425925661. K1096 obviously contains the isogonal conjugate (with respect to P1P2P3) of any of its points. In particular, • X(112)* = X(112)X(511) /\ X(2592)X(2593), SEARCH = 1.45605557616671. • X(31954) = X(1113)* and X(31955) = X(1114)* lie on the line X(6)X(74) and on the parallels at X(1113) and X(1114) to the Brocard axis respectively. • X(2592)* = X(112)X(1113) /\ X(511)X(2592) and X(2593)* = X(112)X(1114) /\ X(511)X(2593).

